| L(s) = 1 | + (0.573 + 0.819i)2-s + (−0.190 − 2.17i)3-s + (−0.342 + 0.939i)4-s + (1.67 − 1.40i)6-s + (−0.115 + 0.431i)7-s + (−0.965 + 0.258i)8-s + (−1.74 + 0.307i)9-s + (2.37 − 4.12i)11-s + (2.11 + 0.565i)12-s + (−2.40 − 0.210i)13-s + (−0.419 + 0.152i)14-s + (−0.766 − 0.642i)16-s + (−0.560 + 0.392i)17-s + (−1.25 − 1.25i)18-s + (2.15 − 3.79i)19-s + ⋯ |
| L(s) = 1 | + (0.405 + 0.579i)2-s + (−0.109 − 1.25i)3-s + (−0.171 + 0.469i)4-s + (0.683 − 0.573i)6-s + (−0.0436 + 0.163i)7-s + (−0.341 + 0.0915i)8-s + (−0.582 + 0.102i)9-s + (0.717 − 1.24i)11-s + (0.609 + 0.163i)12-s + (−0.666 − 0.0582i)13-s + (−0.112 + 0.0408i)14-s + (−0.191 − 0.160i)16-s + (−0.135 + 0.0951i)17-s + (−0.295 − 0.295i)18-s + (0.493 − 0.869i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.200 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27951 - 1.04390i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27951 - 1.04390i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.573 - 0.819i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-2.15 + 3.79i)T \) |
| good | 3 | \( 1 + (0.190 + 2.17i)T + (-2.95 + 0.520i)T^{2} \) |
| 7 | \( 1 + (0.115 - 0.431i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-2.37 + 4.12i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.40 + 0.210i)T + (12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (0.560 - 0.392i)T + (5.81 - 15.9i)T^{2} \) |
| 23 | \( 1 + (0.404 - 0.188i)T + (14.7 - 17.6i)T^{2} \) |
| 29 | \( 1 + (0.892 + 5.06i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-2.27 + 1.31i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.48 - 1.48i)T - 37iT^{2} \) |
| 41 | \( 1 + (0.333 - 0.397i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.94 + 8.46i)T + (-27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (3.02 - 4.32i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (3.42 + 7.35i)T + (-34.0 + 40.6i)T^{2} \) |
| 59 | \( 1 + (-0.0555 + 0.315i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (7.77 + 2.82i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (9.20 + 6.44i)T + (22.9 + 62.9i)T^{2} \) |
| 71 | \( 1 + (-0.879 - 2.41i)T + (-54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-4.32 + 0.378i)T + (71.8 - 12.6i)T^{2} \) |
| 79 | \( 1 + (-12.3 - 10.3i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.838 + 0.224i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-12.7 + 10.7i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-1.72 - 2.46i)T + (-33.1 + 91.1i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.619095700562876739351186352144, −8.780624508154398867323350644993, −7.927526246834797866181192799493, −7.24011733113983145621592573446, −6.41363025397264292592119193956, −5.86040107993777583585285099186, −4.71813259983621368215084651508, −3.45731873513282648467244040251, −2.27876823591755395804716949540, −0.70631728226833540334394327820,
1.67120059088464352506741871933, 3.10651927710579851224349221222, 4.10824326331642104670188119640, 4.66492730217243807445138994959, 5.51520712911645292962289928720, 6.72838797448316061420469378588, 7.67192419645988827752567103998, 9.106705578237321261988825739132, 9.538124976789429647690270597553, 10.28496113895015571671958297030
